Large language models (LLMs) show promise for improving the efficiency of qualitative analysis in large, multi-site health-services research. Yet methodological guidance for LLM integration into qualitative analysis and evidence of their impact on real-world research methods and outcomes remain limited. We developed a model- and task-agnostic framework for designing human-LLM qualitative analysis methods to support diverse analytic aims. Within a multi-site study of diabetes care at Federally Qualified Health Centers (FQHCs), we leveraged the framework to implement human-LLM methods for (1) qualitative synthesis of researcher-generated summaries to produce comparative feedback reports and (2) deductive coding of 167 interview transcripts to refine a practice-transformation intervention. LLM assistance enabled timely feedback to practitioners and the incorporation of large-scale qualitative data to inform theory and practice changes. This work demonstrates how LLMs can be integrated into applied health-services research to enhance efficiency while preserving rigor, offering guidance for continued innovation with LLMs in qualitative research.